Comparative Analysis of the E.D.E.N. Geodesic Topology
How a dual-icosahedral, hexagon-dominant mesh compares to alternative global discretization methods.
The E.D.E.N. planetary engine is built on a specific geodesic discretization: a dual-icosahedral spherical mesh composed primarily of hexagons with twelve pentagons. This structure provides near-uniform area distribution, consistent adjacency, and a natural basis for environmental modeling.
This document compares that topology to other commonly used planetary discretization methods, including latitude–longitude rasters, cubed-sphere grids, planar projections, unstructured meshes, and more.
1. Overview of the E.D.E.N. Topology
The simulation uses:
- Base geometry: Icosahedron
- Subdivision: Recursive refinement of edges, projected to a sphere
- Cell composition: Hexagon-dominant with exactly 12 pentagons
- Adjacency: Mostly 6-neighbor tiles, 5-neighbor at pentagons
- Uniformity: Near-equal-area cells, minimal distortion
- Data model: Tile, edge, and directed-edge fields
This configuration eliminates poles, preserves surface symmetry, and provides a consistent substrate for scalar, vector, and categorical fields across the entire sphere.
2. Evaluation Criteria
The comparison below evaluates each topology using the following attributes:
- Area Uniformity
- Adjacency Regularity
- Geometric Distortion / Singularities
- Hierarchical Refinement
- Handling of Flow / Vector Fields
- Numerical Stability
- Visual and Cognitive Interpretability
- Implementation Complexity
- Compatibility with Earth-data Standards
3. Latitude–Longitude Grids
Latitude–longitude grids divide the planet using equal angular increments in both directions.
Advantages
- Very simple indexing
- Direct compatibility with most global datasets
- Supported by nearly all GIS and climate tools
Limitations
- Severe area distortion: cells shrink dramatically near the poles
- Singularities at the poles disrupt adjacency and flow
- Strong anisotropy: neighbor distances vary widely with latitude
- Complications when modeling vector fields and flows
Comparison
The geodesic topology avoids these issues by providing consistent cell areas, no pole singularities, and uniform adjacency globally.
4. Planar Grids + Map Projections
Planar square or hex grids are often mapped to a sphere using a projection.
Advantages
- Straightforward 2D representation
- Familiar in game design and visualization contexts
Limitations
- All projections introduce distortion (area, shape, or distance)
- Adjacency relationships do not match true spherical geometry
- Physical processes simulated on the projection require correction
Comparison
The geodesic topology simulates directly on the sphere, eliminating projection distortion during computation.
5. Cubed-Sphere / Quad-Sphere Grids
Cubed-sphere grids start with a cube and project its faces onto the sphere, then subdivide them.
Advantages
- No true poles
- More uniform than lat/long
- Simple (i, j, face) indexing
Limitations
- Six face seams introduce subtle anisotropies
- Cell shapes vary from face center to edges
- Degree-4 adjacency is less isotropic for flow and diffusion modeling
Comparison
While widely used in climate models, cubed-sphere grids remain less isotropic than dual-icosahedral geodesic meshes, which offer smoother symmetry and better support for radial flows.
6. Generic Spherical Meshes (Non-Uniform Hex/Tri)
These meshes often originate from arbitrary triangulations or artist-generated geometry.
Advantages
- Flexible generation
- Adaptive meshes are possible
Limitations
- No inherent guarantees about area or neighbor regularity
- Harder to analyze or visualize consistently
- Poor foundation for systematic environmental models
Comparison
The E.D.E.N. topology uses a mathematically controlled mesh with predictable behavior, essential for deterministic simulation.
7. HEALPix (Hierarchical Equal Area Pixelization)
A well-known spherical tessellation used in astrophysics.
Advantages
- Precisely equal-area cells
- Strong existing scientific usage
Limitations
- Not hex-based; cell shapes vary
- Adjacency patterns differ from uniform hex grids
- Less natural for edge-based or directional field modeling
Comparison
The E.D.E.N. topology offers more uniform neighbor relationships and aligns more naturally with tile/edge/directed-edge models.
8. Unstructured / Voronoi / FEM-Style Meshes
Common in high-end scientific simulations requiring flexible element shapes.
Advantages
- Allows adaptive refinement in specific regions
- Highly accurate for targeted scientific problems
Limitations
- Complex neighbor relationships
- Harder to visualize consistently
- Poor match for general-purpose or educational use
- Overhead of arbitrary cell geometry
Comparison
While powerful for specialized scientific workloads, unstructured meshes introduce too much complexity for a general, extensible planetary simulation environment.
9. Planar Hex Grids (2D)
Hex grids offer excellent local properties in 2D.
Advantages
- Uniform area, distance, and adjacency
- Classic structure for cellular systems
Limitations
- Not compatible with spherical topology
- Requires artificial wrap-around (toroidal distortions)
Comparison
The E.D.E.N. mesh extends the advantages of hex grids to a sphere without topological artifacts.
10. Summary of Advantages of the E.D.E.N. Topology
The dual-icosahedral geodesic mesh offers:
- No poles or singularities
- Near-equal-area cells globally
- Consistent 6-neighbor adjacency (except 12 pentagons)
- Smooth global symmetry
- Natural subdivision hierarchy
- Clear tile/edge/directed-edge relationships
- Strong suitability for vector and flow fields
- Clean, interpretable visualization
These characteristics make it particularly well-suited for:
- environmental and climate simulation
- educational clarity
- emergent systems modeling
- scientific visualization
- robust SDK and extension frameworks
The geodesic topology is therefore an ideal foundation for a modern, extensible planetary simulation ecosystem.